Rank Two Quiver Gauge Theory, Graded Connections and Noncommutative Vortices

We consider equivariant dimensional reduction of Yang-Mills theory on K"ahler manifolds of the form M times CP^1 times CP^1. This induces a rank two quiver gauge theory on M which can be formulated as a Yang-Mills theory of graded connections on M. The reduction of the Yang-Mills equations on M times CP^1 times CP^1 induces quiver gauge theory equations on M and quiver vortex equations in the BPS sector. When M is the noncommutative space R_theta^{2n} both BPS and non-BPS solutions are obtained, and interpreted as states of D-branes. Using the graded connection formalism, we assign D0-brane charges in equivariant K-theory to the quiver vortex configurations. Some categorical properties of these quiver brane configurations are also described in terms of the corresponding quiver representations.